Perfect Numbers

Date: 12/07/96 at 23:51:42
From: Ken
Subject: Perfect Number
What are the first 10 perfect numbers? Is there a formula for getting
a perfect number?

Date: 01/26/97 at 16:26:16
From: Doctor Reno
Subject: Re: Perfect Number
Hi, Ken!
Perfect numbers are something that I recently became interested in.
They have a fascinating history in mathematics. My response will be
somewhat detailed due to the richness of the material. With this
information, you will be able to figure out how to find all of the
perfect numbers that you want to find!
By definition, a perfect number is a number which equals the sum of
its factors. In other words, 6 is perfect because the factors of 6
are 1, 2 and 3 and 1 + 2 + 3 = 6. 28 is the next perfect number
because its factors are 1 + 2 + 4 + 7 + 14, whose sum is 28.
Over two thousand years ago, Euclid wrote about constructing perfect
numbers in this way:
"If as many numbers as we please beginning from an unit be set
out continuously in double proportion, until the sum of all
becomes prime, and if the sum multiplied into the last make
some number, the product will be perfect."
This sounds complicated, but if we take it step by step, we can
understand what Euclid was talking about! I will take phrases from
this long sentence and we will explore what is happening. (In case
the ancient lingo is a little confusing, a "unit" is 1, and "double
proportion" means doubling, or multiplying by 2.)
"an unit be set out continuously in double proportion":
1,2,4,8,16,32.... (1 x 2 = 2, 2 x 2 = 4, 4 x 2 = 8, etc)
...and Euclid says we can do this as many times as we please!
"if the sum":
easy! we can do this one!
1 + 2 + 4 + 8 = 16
"until the sum of all becomes prime": simply means that we add until
we get to a prime number!
1 + 2 = 3 That's easy, isn't it!
"the sum multiplied into the last make some number"
Easier than it sounds: simply multiply the sum to the last number:
3 x 2 = 6
"the product will be perfect"
There you have it! 6 is the first perfect number!
Using this method, we can easily find the next perfect number:
1 + 2 + 4 = 7
7 x 4 = 28: the second perfect number!
And the next......
1 + 2 + 4 + 8 = 15....oops! 15 is NOT prime, so we have to go on.
1 + 2 + 4 + 8 + 16 = 31 OK! 31 is prime, so we can use Euclid's
method again.
31 x 16 = 496 The third perfect number.
And the next?
1 + 2 + 4 + 8 + 16 + 32 = 63 63 is NOT PRIME
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 127 is prime (it would help to
have a list of primes, wouldn't it?!)
127 x 64 = 8,128 (a calculator will help, too!)
The fifth perfect number (I've done a little of the work for you):
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 + 4096
= 8191
8191 x 4096 = 33,550,336
We now have the first five perfect numbers:
6
28
496
8128
33,550,336
As you can see, it will be difficult to find the first ten perfect
numbers....they will get very, very large! If you need a table of
prime numbers, I have found one at:
http://www.utm.edu/research/primes/lists/small/10000.txt
Do you see some patterns in these numbers? Many mathematicians
throughout history have been fascinated by the patterns that you also
see! They have played with perfect numbers and come up with a lot of
different ideas about them.
The formula for finding perfect numbers is:
2^n-1(2^n - 1), where (2^n - 1) is prime.
It is not known at this time whether there are an infinite number of
perfect numbers or not.
Here is a web site that gives some history and references for perfect
numbers: MathLand:
http://www.sciencenews.org/sn_arc97/1_25_97/mathland.htm
I hope you have as much fun finding your perfect numbers as I had
helping you with your questions!
-Doctor Reno, The Math Forum